Tunable focusing is a desired property in a wide range of optical imaging and sensing technologies but has tended to require bulky components that cannot be integrated on-chip and have slow actuation speeds. Recently, integration of metasurfaces into electrostatic micro-electromechanical system (MEMS) architectures has shown potential to overcome these challenges but has offered limited out-of-plane displacement range while requiring large voltages. We demonstrate for the first time, to the best of our knowledge, a movable metasurface lens actuated by integrated thin-film PZT MEMS, which has the advantage of offering large displacements at low voltages. An out-of-plane displacement of a metasurface in the range of 7.2 μm is demonstrated under a voltage application of 23 V. This is roughly twice the displacement at a quarter of the voltage of state of the art electrostatic out-of-plane actuation of metasurfaces. Using this tunability, we demonstrate a varifocal lens doublet with a focal shift of the order of 250 μm at the wavelength 1.55 μm. The thin-film PZT MEMS-metasurface is a promising platform for miniaturized varifocal components.